Download Introduction to Integers Get the Point?

Introduction to Integers
Get the Point?
ACTIVITY
1.8
SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/
Retell, Create Representations, Quickwrite, Self Revision/
Peer Revision
My Notes
Ms. Martinez has a point system in her classroom. Students earn
points for participation, doing homework, using teamwork, and
so on. However, students lose points for talking or not completing
homework or class work.
Ms. Martinez tells the class that at the end of the year each
student in the group with the most points will receive a book or
DVD. She assigns a letter to each student so she can easily track
point totals. One student is A, the next is B, and so on.
1. T his table shows each student’s total points at the end of the
week. Your teacher will assign you a letter.
A
−3
B
3
C
8
D
−1
E
0
F
−5
G
−6
H
10
I
7
J
−4
K
1
L
2
M
−3
N
−2
O
12
P
1
Q
−7
R
2
S
−1
T
6
U
−4
V
−1
W
9
X
3
a. Write the total points and the letter assigned to you on a
sticky note. Then post it on the class number line.
© 2010 College Board. All rights reserved.
b. Copy the letters from the class record on this number line.
–6 –4 –2
0
2
4
6
8
10 12
c. In terms of points, what do the numbers to the right of zero
represent?
d. What do the numbers to the lef t of zero represent?
e. Describe how you knew where to place your number on the
number line.
f. Student E was in class for only 2 days during the week. On
the first day, E was awarded points, and on the second day,
E lost points. Explain why E’s score is zero.
Unit 1 • Number Concepts
45
ACTIVITY 1.8
Introduction to Integers
continued
Get the Point?
SUGGESTED LEARNING STRATEGIES: Marking the Text,
Create Representations, Think/Pair/Share
My Notes
CONNECT TO HISTORY
The Common Era is the calendar
system now used throughout the
world. This system is like a number line because the year numbers
increase as time moves on. The
label CE can be used for these
years. For example, the first year
of the twenty-first century could
be written 2001 CE.
2. We have seen how negative numbers can be used to represent
points lost by the students. Name at least three other uses for
negative numbers in real life.
Ms. Martinez sometimes assigns cooperative learning groups. She
assigns each group member a role based on his or her total points.
T he roles are reporter (lowest total), recorder (next to lowest total),
facilitator (next to highest total), and timekeeper (highest total).
3. Use the number lines.
a. Order the points for the members in each group from lowest
to highest.
Group 1:
–4 –2
B
3
0
C
8
2
4
D
−1
6
8
Group 3:
I
7
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SpringBoard® Mathematics with Meaning™ Level 1
–6 –4 –2
F
−5
0
G
−6
2
4
H
10
6
8
10
Group 4:
J
−4
K
1
L
2
Group 5:
Q
−7
E
0
M
−3
N
−2
O
12
P
1
V
−1
W
9
X
3
Group 6:
R
2
S
−1
T
6
U
−4
© 2010 College Board. All rights reserved.
A
−3
Group 2:
Introduction to Integers
ACTIVITY 1.8
continued
Get the Point?
SUGGESTED LEARNING STRATEGIES: Create Representations,
Marking the Text, Summarize/Paraphrase/Retell
My Notes
b. Use your work in Part a to determine who will have each
role. Record the letters for each student in the table.
Reporter
Recorder
Facilitator Time Keeper
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
4. Look at Students A and B.
a. How many points does A need to earn to have a total of 0?
b. How many points does B have to lose to have a total of 0?
© 2010 College Board. All rights reserved.
c. What do you notice about the distances of their point totals
from zero?
Numbers that are the same distance from zero and are on
dif ferent sides of zero on a number line, such as −3 and 3, are
called opposites. Absolute value is the distance from zero and
is represented with bars: |−3| = 3 and |3| = 3. Absolute value is
always positive because distance is always positive.
5. Now f ind C’s and D’s distances from zero.
6. What is the absolute value of zero?
ACADEMIC VOCABULARY
The absolute value of a
number is the distance of
the number from zero on
a number line. Distance or
absolute value is always
positive. For example, the
absolute value of both –6
and 6 is 6.
Absolute value can be used to compare and order positive and
negative numbers. The negative number that is the greatest distance
from zero is the smallest. |−98| = 98 and the |−90| = 90. Therefore,
−98 is further left from zero than −90 is, so −98 is less than −90.
7. Use this method to compare each pair of negative numbers.
−15
−21
−392
−390
−2,840
−2,841
Unit 1 • Number Concepts
47
ACTIVITY 1.8
Introduction to Integers
continued
Get the Point?
SUGGESTED LEARNING STRATEGIES: Summarize/
Paraphrase/Retell, Identify a Subtask
My Notes
ACADEMIC VOCABULARY
Integers are the natural
numbers, their opposites,
and zero.
The opposite of 0 is 0.
The number lines below give visual representations of the integers.
Notice that zero is the only integer that is neither positive nor
negative.
Opposites
0 +1 +2 +3 +4 +5 +6
–6 –5 –4 –3 –2 –1
Opposite of Natural Numbers
Natural Numbers
WRITING MATH
0 +1 +2 +3 +4 +5 +6
–6 –5 –4 –3 –2 –1
Place a negative sign in front of a
number to indicate its opposite.
Negative Integers
Positive Integers
The opposite of 4 is -4.
MATH TERMS
T he cooperative groups will f ind their totals to determine which
group has the most points at this time.
8. Group 1 uses a number line to f ind their total.
Most mathematicians use Z to
refer to the set of integers. This
is because in German, the word
Zahl means “number.”
A
−3
–3 –2 –1
WRITING MATH
To avoid confusion, use
parentheses around a negative number that follows an
operation symbol.
B
3
0
1
C
8
2
3
D
−1
4
5
6
7
8
To add with a number line, start at the f irst number. T hen move to
the right to add a positive number, or to the lef t to add a negative
number.
a. Add A and B, or −3 + 3: Put your pencil at −3 and move it
to the right 3 places to add 3.
b. Add C and D, or 8 + (−1): Put your pencil on 8 and move it
to the lef t 1 place to add −1.
c. Combine the sums of Parts a and b.
48
SpringBoard® Mathematics with Meaning™ Level 1
© 2010 College Board. All rights reserved.
The opposite of -4 is –(-4) = 4.
Introduction to Integers
ACTIVITY 1.8
continued
Get the Point?
SUGGESTED LEARNING STRATEGIES: Quickwrite, Think/Pair/
Share, Create Representations, Role Play, Use Manipulatives
My Notes
9. What happens when you add a number and its opposite, for
example, 3 and −3?
10. A number and its opposite are called additive inverses.
a. Why do you think they are also called a zero pair?
b. Write 2 more zero pairs.
ACADEMIC VOCABULARY
A number and its opposite
are called additive inverses.
The sum of a number and its
additive inverse is zero.
11. Use one number line to f ind the total points for group 2.
E
0
F
G
−5 −6
–12 –10 –8 –6 –4 –2
0
2
H
10
4
Remember, zero pairs have a sum
of zero (-1 + 1 = 0), so they are
eliminated.
6
8
10 12
Group 3 decides to add their points using positive and negative
counters.
© 2010 College Board. All rights reserved.
I
7
A positive counter
J
−4
K
1
J
2
is +1. A negative counter
is −1.
To add I and J, f irst take 7 positive counters and 4 negative
counters.
Next, combine the counters in zero pairs.
3 positive counters remain, so 7 + (−4) = 3
12. What is the point total of Group 3?
Unit 1 • Number Concepts
49
ACTIVITY 1.8
Introduction to Integers
continued
Get the Point?
My Notes
SUGGESTED LEARNING STRATEGIES: Use Manipulatives,
Look for a Pattern, Group Discussion, Self Revision/Peer
Revision, Identify a Subtask
13. Use positive and negative counters to add the points of the
students in Group 4.
M
-3
N
-2
O
12
P
1
a. Draw counters to show -3 + (-2).
b. What is the sum for students O and P?
c. Draw counters to add your answers to Parts a and b.
3+5=8
-2 + -3 = -5
-3 + -5 = -8
4 + 9 = 13
2+3=5
-4 + -9 = -13
14. Look for a pattern and write a generalization for adding
integers with the same signs.
Next, they consider sums of integers with different signs:
-2 + 5 = 3
7 + -11 = -4
4 + -3 = 1
-8 + 1 = -7
15. Look for a pattern and write a generalization for adding
integers with different signs.
50
SpringBoard® Mathematics with Meaning™ Level 1
© 2010 College Board. All rights reserved.
Group 5 looks at number relationships in order to find a way to add
integers without the help of number lines or counters.
First, they look at the sums of integers that have the same signs:
Introduction to Integers
ACTIVITY 1.8
continued
Get the Point?
SUGGESTED LEARNING STRATEGIES: : Identify a Subtask,
Create Representations, Role Play
My Notes
16. Use these generalizations to find the total points for the
students in Group 5.
Q
-7
R
2
S
-1
T
6
a. Q + R -7 + 2 =
b. (Q + R) + S
c. Find the sum for all four students.
17. Use your generalizations to find the total points for the
students in Group 6.
U
-4
V
-1
W
9
X
3
18. Compile the total group points in the table below.
© 2010 College Board. All rights reserved.
G1
G2
G3
G4
G5
G6
19. Which group has the most points?
You can evaluate the numerical expression 8 - (-1) using
positive and negative counters.
MATH TERMS
A numerical expression is a
number, or a combination of
numbers and operations.
• Begin with 8 positive counters (+8):
• You must subtract a negative counter (-1), but there are none.
So, add a zero pair, 1 positive and 1 negative.
+
• Now, subtract the negative counter, leaving 9 positive counters.
So, 8 - (-1) = 9.
+
Thus, subtracting a negative 1 is like adding a positive 1.
8 - (-1) = 8 + 1
Unit 1 • Number Concepts
51
ACTIVITY 1.8
Introduction to Integers
continued
Get the Point?
SUGGESTED LEARNING STRATEGIES: Use Manipulatives,
Identify a Subtask, Think/Pair/Share
My Notes
20. Draw counters to find the difference: 3 - (-4). Do your work
in the My Notes space.
a. What type of counters do you need to start, and how many
do you need?
b. What type of counters do you need to subtract, and
how many?
c. Notice that you do not have the counters you need to
subtract. What can you add to give you the counters you
need without changing the starting value?
d. Cross out four negative counters (subtract - 4). Remaining
counters: = 3 - (-4) =
21. Complete this statement to show how to compute 3 - (-4)
by adding the opposite.
22. Find the difference two ways.
a. -5 - 7
Use counters:
Add the opposite:
b. -2 - (-3)
Use counters:
Add the opposite:
c. 8 - (-6)
Use counters:
Add the opposite:
52
SpringBoard® Mathematics with Meaning™ Level 1
.
© 2010 College Board. All rights reserved.
Instead of subtracting -4, add its opposite, 4.
The addition expression is
, so 3 - (-4) =
Introduction to Integers
ACTIVITY 1.8
continued
Get the Point?
CHECK YOUR UNDERSTANDING
Write your answers
answers on
on notebook
notebook paper.
paper.Show your work.
4. Evaluate each expression.
Show your work.
a. -3 + 9
b. -5 + (-7)
1. Write an integer to represent each situation.
c. -12 + 6
d. -24 - 11
a. 10 yard loss in football
e. -13 - (-8)
b. Earn $25 at work
5. During their possession, a football team
gained 5 yards, lost 8 yards, lost another
2 yards, then gained 45 yards. What were
the total yards gained or lost?
c. 2 degrees below zero
d. Elevation of 850 feet above sea level
e. Change in score after an inning with
no runs
f. The opposite of losing 50 points in a game
2. Evaluate each expression.
1
c. - __
2
3. The following table shows the high and
low temperatures of 5 consecutive days in
February in North Pole, Alaska.
© 2010 College Board. All rights reserved.
a. |54|
High
Low
Mon
1
-13
b. |-11|
Tues
-29
-45
Wed
-27
-54
|
Thurs
5
-2
f. 31 - (-10)
|
Fri
7
1
a. Order the high temperatures from warmest
to coldest over this five-day period.
6. In North America the highest elevation
is Denali in Alaska at 20,320 feet above
sea level and the lowest elevation is Death
Valley in California at 282 feet below sea
level. Write and evaluate an expression
with integers to find the difference between
the elevations.
7. On winter morning, the temperature fell
below -6°C. What does this temperature
mean in terms of 0˚C?
8. MATHEMATICAL Using a number line,
R E F L E C T I O N explain how you can
order integers.
b. Is the order of days from warmest to coldest
daily low temperatures the same as for the
daily high temperatures? Explain.
Unit 1 • Number Concepts
53